First-crossing Time Research Articles

Adaptive First-Crossing Approach for Life-Cycle Reliability Analysis

Abstract Life-cycle reliability analysis can effectively estimate and present the changes in the state of safety for structures under dynamic uncertainties during their lifecycle. The first-crossing approach is an efficient way to evaluate time-variant reliability-based on the probabilistic characteristics of the first-crossing time point (FCTP). However, the FCTP model has a number of critical challenges, such as computational accuracy. This paper proposes an adaptive first-crossing approach for the time-varying reliability of structures over their whole lifecycle, which can provide a tool for cycle-life reliability analysis and design. The response surface of FCTP regarding input variables is first estimated by performing support vector regression. Furthermore, the adaptive learning algorithm for training support vector regression is developed by integrating the uniform design and the central moments of the surrogate model. Then, the convergence condition, which combines the raw moments and entropy of the first-crossing probability distribution function (PDF), is constructed to build the optimal first-crossing surrogate model. Finally, the first-crossing PDF is solved using the adaptive kernel density estimation to obtain the time-variant reliability trend during the whole lifecycle. Examples are demonstrated to specify the proposed method in applications.

Encounter-based approach to the escape problem.

We revise the encounter-based approach to imperfect diffusion-controlled reactions, which employs the statistics of encounters between a diffusing particle and the reactive region to implement surface reactions. We extend this approach to deal with a more general setting, in which the reactive region is surrounded by a reflecting boundary with an escape region. We derive a spectral expansion for the full propagator and investigate the behavior and probabilistic interpretations of the associated probability flux density. In particular, we obtain the joint probability density of the escape time and the number of encounters with the reactive region before escape, and the probability density of the first-crossing time of a prescribed number of encounters. We briefly discuss generalizations of the conventional Poissonian-type surface reaction mechanism described by Robin boundary condition and potential applications of this formalism in chemistry and biophysics.

Statistics of diffusive encounters with a small target: three complementary approaches

Diffusive search for a static target is a common problem in statistical physics with numerous applications in chemistry and biology. We look at this problem from a different perspective and investigate the statistics of encounters between the diffusing particle and the target. While an exact solution of this problem was recently derived in the form of a spectral expansion over the eigenbasis of the Dirichlet-to-Neumann operator, the latter is generally difficult to access for an arbitrary target. In this paper, we present three complementary approaches to approximate the probability density of the rescaled number of encounters with a small target in a bounded confining domain. In particular, we derive a simple fully explicit approximation, which depends only on a few geometric characteristics such as the surface area and the harmonic capacity of the target, and the volume of the confining domain. We discuss the advantages and limitations of three approaches and check their accuracy. We also deduce an explicit approximation for the distribution of the first-crossing time, at which the number of encounters exceeds a prescribed threshold. Its relations to common first-passage time problems are discussed.

Statistics of boundary encounters by a particle diffusing outside a compact planar domain

We consider a particle diffusing outside a compact planar set and investigate its boundary local time ℓ t , i.e., the rescaled number of encounters between the particle and the boundary up to time t. In the case of a disk, this is also the (rescaled) number of encounters of two diffusing circular particles in the plane. For that case, we derive explicit integral representations for the probability density of the boundary local time ℓ t and for the probability density of the first-crossing time of a given threshold by ℓ t . The latter density is shown to exhibit a very slow long-time decay due to extremely long diffusive excursions between encounters. We briefly discuss some practical consequences of this behavior for applications in chemical physics and biology.

Time-variant reliability analysis via approximation of the first-crossing PDF

Time-variant reliability analysis can effectively estimate the safe state of structures under dynamic uncertainties during their lifecycle. However, one of its key challenging issues is computational efficiency. To improve the efficiency, this paper develops an approximation of the first-crossing probability density function (PDF) method, termed AFC-PDFM, based on the first-crossing theory. The response surface of the first-crossing time point (FCTP) about input variables is first estimated by using the moving least squares method (MLSM) and uniform design. Furthermore, an iterative algorithm is developed to compute the origin moments of the FCTP based on the MLSM response surface. Finally, the PDF of the FCTP is obtained by combining the origin moments and the maximum entropy method to avoid direct calculations of the first-crossing rate, which is often intractable, and then the time-variant reliability is estimated. Three examples are implemented to testify and validate the effectiveness of the proposed method and potential applications to engineering problems.

Some Results on the Telegraph Process Confined by Two Non-Standard Boundaries

We analyze the one-dimensional telegraph random process confined by two boundaries, 0 and H > 0. The process experiences hard reflection at the boundaries (with random switching to full absorption). Namely, when the process hits the origin (the threshold H) it is either absorbed, with probability α, or reflected upwards (downwards), with probability 1 − α, for 0 < α < 1. We provide various results on the expected values of the renewal cycles and of the absorption time. The adopted approach is based on the analysis of the first-crossing times of a suitable compound Poisson process through linear boundaries. Our analysis includes also some comparisons between suitable stopping times of the considered telegraph process and of the corresponding diffusion process obtained under the classical Kac’s scaling conditions.

First-Crossing Problem of Weakly Coupled Strongly Nonlinear Oscillators Subject to a Weak Harmonic Excitation and Gaussian White Noises

We study first-crossing problem of two-degrees-of-freedom (2DOF) strongly nonlinear mechanical oscillators analytically. The excitation is the combination of a deterministic harmonic function and Gaussian white noises (GWNs). The generalized harmonic function is used to approximate the solutions of the original equations. Four cases are studied in terms of the types of resonance (internal or external or both). For each case, the method of stochastic averaging is used and the stochastically averaged Itô equations are obtained. A backward Kolmogorov (BK) equation is set up to yield the failure probability and a Pontryagin equation is set up to yield average first-crossing time (AFCT). A 2DOF Duffing-van der Pol oscillator is chosen as an illustrative example to demonstrate the effectiveness of the analytical method. Numerically analytical solutions are obtained and validated by digital simulation. It is shown that the proposed method has high efficiency while still maintaining satisfactory accuracy.

Compound Poisson Process with a Poisson Subordinator

A compound Poisson process whose randomized time is an independent Poisson process is called a compound Poisson process with Poisson subordinator. We provide its probability distribution, which is expressed in terms of the Bell polynomials, and investigate in detail both the special cases in which the compound Poisson process has exponential jumps and normal jumps. Then for the iterated Poisson process we discuss some properties and provide convergence results to a Poisson process. The first-crossing time problem for the iterated Poisson process is finally tackled in the cases of (i) a decreasing and constant boundary, where we provide some closed-form results, and (ii) a linearly increasing boundary, where we propose an iterative procedure to compute the first-crossing time density and survival functions.

Compound Poisson Process with a Poisson Subordinator

A compound Poisson process whose randomized time is an independent Poisson process is called a compound Poisson process with Poisson subordinator. We provide its probability distribution, which is expressed in terms of the Bell polynomials, and investigate in detail both the special cases in which the compound Poisson process has exponential jumps and normal jumps. Then for the iterated Poisson process we discuss some properties and provide convergence results to a Poisson process. The first-crossing time problem for the iterated Poisson process is finally tackled in the cases of (i) a decreasing and constant boundary, where we provide some closed-form results, and (ii) a linearly increasing boundary, where we propose an iterative procedure to compute the first-crossing time density and survival functions.

Intermediate-level crossings of a first-passage path

We investigate some simple and surprising properties of a one-dimensional Brownian trajectory with diffusion coefficient D that starts at the origin and: (i) is at X at time T, or (ii) first reaches X at time T. We determine the most likely location of the first-passage trajectory from (0, 0) to (X, T) and its distribution at any intermediate time t < T. A first-passage path typically starts out by being repelled from its final location when X2/DT ≪ 1. We also determine the distribution of times when the trajectory first crosses and last crosses an arbitrary intermediate position x < X. The distribution of first-crossing times may be unimodal or bimodal, depending on whether X2/DT ≪ 1 or X2/DT ≫ 1. The form of the first-crossing probability in the bimodal regime is qualitatively similar to, but more singular than, the well-known arcsine law.

Investigating the Distribution of the First-Crossing Area of a Diffusion Process with Jumps Over a Threshold

For a given barrier $S$ and a one-dimensional jump-diffusion process $X(t),$ starting from $x<S,$ we study the probability distribution of the integral $A_S(x)= \int_0 ^ {\tau_S(x)}X(t) \ dt$ determined by $X(t)$ till its first-crossing time $\tau_S(x)$ over $S.$ In particular, we show that the Laplace transform and the moments of $A_S(x)$ are solutions to certain partial differential-difference equations with outer conditions. The distribution of the minimum of $X(t)$ in $[0, \tau_S(x)]$ is also studied. Thus, we extend the results of a previous paper by the author, concerning the area swept out by $X(t)$ till its first-passage below zero. Some explicit examples are reported, regarding diffusions with and without jumps.

First-crossing and ballot-type results for some nonstationary sequences

In this paper we consider the problem of first-crossing from above for a partial sums process m+S t , t ≥ 1, with the diagonal line when the random variables X t , t ≥ 1, are independent but satisfying nonstationary laws. Specifically, the distributions of all the X t s belong to a common parametric family of arithmetic distributions, and this family of laws is assumed to be stable by convolution. The key result is that the first-crossing time distribution and the associated ballot-type formula rely on an underlying polynomial structure, called the generalized Abel-Gontcharoff structure. In practice, this property advantageously provides simple and efficient recursions for the numerical evaluation of the probabilities of interest. Several applications are then presented, for constant and variable parameters.

First-crossing and ballot-type results for some nonstationary sequences

In this paper we consider the problem of first-crossing from above for a partial sums process m+St, t ≥ 1, with the diagonal line when the random variables Xt, t ≥ 1, are independent but satisfying nonstationary laws. Specifically, the distributions of all the Xts belong to a common parametric family of arithmetic distributions, and this family of laws is assumed to be stable by convolution. The key result is that the first-crossing time distribution and the associated ballot-type formula rely on an underlying polynomial structure, called the generalized Abel-Gontcharoff structure. In practice, this property advantageously provides simple and efficient recursions for the numerical evaluation of the probabilities of interest. Several applications are then presented, for constant and variable parameters.

Generalized integrated telegraph processes and the distribution of related stopping times

Let {X(t), V(t), t ≥ 0} be a telegraph process, with V(0+) = 1. The distribution of X(t) is derived for the general case of an alternating renewal process, describing the length of time a particle is moving to the right or to the left. The distributions of the first-crossing times of given levels a and −a are studied for M/G and for G/M processes.

Generalized integrated telegraph processes and the distribution of related stopping times

Let {X(t),V(t),t≥ 0} be a telegraph process, withV(0+) = 1. The distribution ofX(t) is derived for the general case of an alternating renewal process, describing the length of time a particle is moving to the right or to the left. The distributions of the first-crossing times of given levelsaand −aare studied for M/G and for G/M processes.

First-Passage Problems in Spatially Extended Systems

First-passage problems are related to such questions as the statistics of first-crossing times (across a boundary, or set of boundaries) of a stochastic processx(t)e.g. a random walker. In the “spatially extended” generalization, the single variablex(t)is replaced by a fieldf (xt), with specified dynamics, and one considers the first passage of f at a given pointx.Such problems are, in general, non-trivial even for rather simple processes such as diffusion. This class of problems will be introduced through coarsening systems and reaction-diffusion problems. Analytic results will be presented for the asymptotics of the one-dimensional two-species annihilation process \(A + B \to 0, \) with different initial densities. The asymptotic decay of the density of the minority species maps onto the survival probabilityP(t)of a diffusing particle moving in an infinite sea of diffusing traps. We show that \(Pt\sim \exp \left( { - \lambda \rho {{{\left[ {Dt} \right]}}^{{1/2}}}} \right), \) wherepis the trap densityDthe trap diffusion constant and \(\lambda = 4/\sqrt {^{\pi ,} } \) independent of the particle diffusion constant. The corresponding results for general dimensions d ≤ 2will also be presented.

Markov chain approximation methods in a class of level-crossing problems

Level-crossing problems are encountered in a number of stochastic processes such as queueing, dams, inventory and in the cumulative sum (CUSUM) charts of statistical quality control. Approximate computational methods, using Markov chains to determine expected first-crossing times in CUSUM charts, are reviewed and extended to more general stochastic processes, and examples are given. Formulas are presented when the underlying processes can be represented or approximated by either normal distributions or by generalized hyperexponential distributions.

The moment generating function of the stopping time for a linearly stopped poisson process

First-crossing times of linear boundaries by Poisson processes are considered. In particular, the moment generating functions of stopping times of the form are examined, where is a homogeneous Poisson counting process and . Exact and asymptotic (in B) expressions are obtained by analyzing certain delay-differential equations derived for this moment generating function by Dvoretsky, Kiefer and Wolfowitz. Further results are obtained for the case of a single, lower boundary by exploiting results of Zacks who gives a series representation for the stopping-time distribution in this case. These results are used to obtain a closed-form expression for the cumulants of the stopping time in this latter case

On a symmetry-based constructive approach to probability densities for two-dimensional diffusion processes

The method earlier introduced for one-dimensional diffusion processes [6] is extended to obtain closed form expressions for the transition p.d.f.'s of two-dimensional diffusion processes in the presence of absorbing boundaries and for the first-crossing time p.d.f.'s through such boundaries. Use of such a method is finally made to analyse a two-dimensional linear process.

On a symmetry-based constructive approach to probability densities for two-dimensional diffusion processes

The method earlier introduced for one-dimensional diffusion processes [6] is extended to obtain closed form expressions for the transition p.d.f.'s of two-dimensional diffusion processes in the presence of absorbing boundaries and for the first-crossing time p.d.f.'s through such boundaries. Use of such a method is finally made to analyse a two-dimensional linear process.